Problem 102 - College Algebra

Question

Simplify by reducing the index of the radical. $$\sqrt[4]{7^{2}}$$

Step-by-Step Solution

Answer

The simplified radical expression for \( \sqrt[4]{7^{2}} \) is \( \sqrt{7} \).

Step 1: Rewrite the Power as a Fraction

Rewrite the power as a fraction. Here, express \( \sqrt[4]{7^{2}} \) as \( 7^{2/4} \). The numerator of the fraction is the power raised on the base number and the denominator is the root of the given expression.

Step 2: Simplify the Fraction

Simplify the fraction. Here, the fraction \( 2/4 \) simplifies to \( 1/2 \). So, the expression becomes \( 7^{1/2} \).

Step 3: Convert the Fraction Power to Radical Form

Convert the fractional power back to radical form. Here, the denominator \( 2 \) becomes the index of the radical and the numerator \( 1 \) becomes the power of \( 7 \) in the radical expression. Therefore, the simplified expression becomes \( \sqrt{7} \).